Linear Equations and Inequality Question?

[JasperKie]

You are to take a mathematics achievement test. Two days before the test you receive the following instructions concerning the point values of the two kinds of question, and how many of each you must answer.
i. There are 10 questions worth 7 points each, and 16 questions worth 5 points each.
ii. You can receive credit for a maximum of 20 questions. Any others you answer will not be scored.
iii. To receive any credit at all, you must answer at least 5 questions.
iv. The number of 5-point questions you answer must be no more than twice the number of 7-point questions.
v. The number of 5-point questions you answer must be more than 1/2 the quantity (number of 7-point questions minus 5).

I managed to get the equations
x≤10
y≤16
x+y≤20
x+y≥5
y≤2x
y>x-5
But after that, I'm lost. Do I just create a graph? Is there a way to get the answers using equations?

 

[tkhunny]

#1 - Define your terms. What are 'x' and 'y'? You can't just throw them out there.
#2 - You seem to be missing the "1/2" in part 'v'.
#3 - Sure. Graph away.
#4 - You may also need x ≥ 0 and y ≥ 0.
#5 - Is there an answer? Is there ANY INTEGER solution to all the inequalities simultaneously?

 

[Dr.Peterson]

You must have omitted part of the problem; you didn't say what it tells you to do!

 

[JasperKie]

You must have omitted part of the problem; you didn't say what it tells you to do!

Oh my you're right! My bad.
a. What are the optimum numbers of 5- and 7-point questions to answer in order to maximize your score?
b. What is the maximum feasible score?
c. What is the minimum feasible score, assuming that you satisfy all of the above requirements, and answer each question correctly?

 

[JasperKie]

#1 - Define your terms. What are 'x' and 'y'? You can't just throw them out there.
#2 - You seem to be missing the "1/2" in part 'v'.
#3 - Sure. Graph away.
#4 - You may also need x ≥ 0 and y ≥ 0.
#5 - Is there an answer? Is there ANY INTEGER solution to all the inequalities simultaneously?

Right, x are questions worth 7 points, and y are questions worth 5.
I thought y>x-5 accounted for constraint v.
And I messed up the post because I didn't include what is asked. Let's start off with

"what are the optimum numbers of 5- and 7-point questions to answer in order to maximize your score?"

I will graph it, but that can be inaccurate and I feel there must be an equation way to get the values. I can recall using system of equations to solve these types of problems but I'm not sure how that'll work here, with all the greater's than and less than's. Is there a way to do this using system of equations?

 

[Steven G]

I know that you did NOT mean to be rude, but seriously just because you know what your variables mean does not mean that others will. So please define your variables.

 

[Steven G]

In V what does quantity mean? Also what is the meaning of what is in the parenthesis?

If quantity means the number of questions answered, then y>.5(x+y) which is equivalent to y>x

 

[JasperKie]

I know that you did NOT mean to be rude, but seriously just because you know what your variables mean does not mean that others will. So please define your variables.

My bad, I wish there was a way to edit it now. x are 7 point questions, and y are 5 point questions

 

[JasperKie]

In V what does quantity mean? Also what is the meaning of what is in the parenthesis?

If quantity means the number of questions answered, then y>.5(x+y) which is equivalent to y>x

The quantity is the amount of total questions, which you want to be 20. The parentheses is essentially the equation for the constraint written in words. The exact equation is y>x-5, which works because x cannot exceed 10. Sorry if it is confusing, as I myself find me being stumped all the time.

 

[JasperKie]

Right, x are questions worth 7 points, and y are questions worth 5.
I thought y>x-5 accounted for constraint v.
And I messed up the post because I didn't include what is asked. Let's start off with

"what are the optimum numbers of 5- and 7-point questions to answer in order to maximize your score?"

I will graph it, but that can be inaccurate and I feel there must be an equation way to get the values. I can recall using system of equations to solve these types of problems but I'm not sure how that'll work here, with all the greater's than and less than's. Is there a way to do this using system of equations?

After graphing, I found that the answer is (9,11), 9 seven point questions, and 11 five point questions. I found this method to be ineffective as my lines weren't clear and it's hard to make it so. Is this the answer?

 

[Steven G]

Oh my you're right! My bad.
a. What are the optimum numbers of 5- and 7-point questions to answer in order to maximize your score?
b. What is the maximum feasible score?
c. What is the minimum feasible score, assuming that you satisfy all of the above requirements, and answer each question correctly?

Oh my you're right! My bad.
a. What are the optimum numbers of 5- and 7-point questions to answer in order to maximize your score?
b. What is the maximum feasible score?
c. What is the minimum feasible score, assuming that you satisfy all of the above requirements, and answer each question correctly?

x≤10
y≤16
x+y≤20
x+y≥5
y≤2x
y>x-5
I would try trial and error to answer parts a and b. Actually they are really the same question.
The first 4 inequalities listed can easily be satisfied. If those were the only 4 inequalities then the answer to part a would be to answer all 10 7-point problems and 10 5-point problems. So x=10 and y=10
Does this satisfy y≤2x? How about y>x-5? If yes, then you are done. If not, then adjust x and y to satisfy the last two inqualities while trying to keep the score maximum.

For part c I would try the same trial and error technique. Letting x=0 and y=5 satisfies the 1st 4 inequalities while yielding the minimum score. Does x=0 and y=5 satisfy the last two inequalities? Clearly y≤2x is not true. So how do you handle this? You must increase x! So will x=1 work, if not will x=2 work? etc
Note that I am not graphing anything. Graphing does work as well

 

[Steven G]

After graphing, I found that the answer is (9,11), 9 seven point questions, and 11 five point questions. I found this method to be ineffective as my lines weren't clear and it's hard to make it so. Is this the answer?

I do not think that is the answer. Also it does not matter if it is correct as you are not sure if it is correct.
Some students for each inequality shade in the good regions and then look for overlapping regions which can be hard to see. I would advise you to shade in the bad regions and then the feasible solution space will be the blank space which is easily seen.
Remember, as already noted, that you do have x>0 and y>0.

 

[Steven G]

I can recall using system of equations to solve these types of problems but I'm not sure how that'll work here, with all the greater's than and less than's. Is there a way to do this using system of equations?

An equation has an equal sign, so you can solve a system of equations as you do not have any equations.